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# 1. Definition & Example

A

**signed permutation**of**size**$n$ is an bijection $\sigma$ of $\{\pm 1,\ldots,\pm n\}$ such that $\sigma(-i) = -\sigma(i)$.We usually denote a signed permutation in

*one-line notation*. This is given by $\pi = [\pi(1),\ldots,\pi(n)]$. E.g., $\pi = [5,-4,2,-3,-1]$ says that

$$\pi(1)=5,\pi(2)=-4,\pi(3)=2,\pi(4)=-3,\pi(5)=-1.$$

the 8 Signed permutations of size 2 | |||||||

[1,2] |
[1,-2] |
[-1,2] |
[-1,-2] |
[2,1] |
[2,-1] |
[-2,1] |
[-2,-1] |

There are $2^n\cdot n! = 2^n \cdot 1 \cdot 2 \cdot 3 \cdots n$ signed permutations of size $n$, see A000165.

# 2. Additional information

The group of signed permutations of size $n$is the Coxeter group of type $B_n$. It is the group of symmetries of a regular hypercube and also known under the name

*hyperoctahedral group*.

# 3. Properties

TBA

# 4. Remarks

TBA

# 5. References

# 6. Sage examples

# 7. Technical information for database usage

- Signed permutations are graded by size.
- The database contains all signed permutations of size at most 5.